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<h2 class="hd hd-2 unit-title">Introduction to Matrices</h2>
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<p>
We have practiced writing equations of motion for coupled systems, and now we are prepared to solve them. In order to do so, it is much simpler to express the system of equations in a notation using matrices and vectors. </p><p>
A matrix is a rectangular array of numbers or other mathematical objects. Although matrices can come in any shape, we will deal exclusively with square ones with the same number of rows and columns, for example: </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\begin{pmatrix} a & & b \\ c & & d \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><p>
In this context, a "vector" is not a single quantity with magnitude and direction (for example, [mathjaxinline]\vec{r}[/mathjaxinline] used to denote the position of a single object in mechanics) but instead is used to denote a single attribute of multiple objects. In particular, [mathjaxinline]\textbf{X}[/mathjaxinline] and [mathjaxinline]\ddot{\textbf{X}}[/mathjaxinline] will be used to denote the position and acceleration of a set of objects. For example, for three objects, we have: </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\textbf{X}= \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p>
and </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\ddot{\textbf{X}}= \begin{pmatrix} \ddot{x}_1 \\ \ddot{x}_2 \\ \ddot{x}_3 \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><p>
Before continuing with our physics examples, we will first review some general properties of matrices and vectors, including: addition and subtraction, multiplication, determinants, and inverse operations. </p>
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<h2 class="hd hd-2 unit-title">L7Q1: Matrix Algebra Review: Addition and Subtraction</h2>
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Adding and Subtracting Matrices
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<p>
Addition and subtraction of matrices are performed on an element-wise basis: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\begin{pmatrix} a &amp; &amp; b \\ c &amp; &amp; d \end{pmatrix}+ \begin{pmatrix} e &amp; &amp; f \\ g &amp; &amp; h \end{pmatrix}= \begin{pmatrix} (a+e) &amp; &amp; (b+f) \\ (c+g) &amp; &amp; (d+h) \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>1</span>)</span>
</td>
</tr>
</table>
<p>
Properties of addition and subtraction include: </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A} + \textbf{B} = \textbf{B} + \textbf{A}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>2</span>)</span>
</td>
</tr>
</table>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax](\textbf{A} + \textbf{B}) + \textbf{C} = \textbf{A} + (\textbf{B} + \textbf{C})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>3</span>)</span>
</td>
</tr>
</table>
<table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A} + 0 = \textbf{A}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>4</span>)</span>
</td>
</tr>
</table>
<p>
Similarly for vectors: </p>
<table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\begin{pmatrix} a \\ b \end{pmatrix}+ \begin{pmatrix} c \\ d \end{pmatrix}= \begin{pmatrix} (a+c) \\ (b+d) \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>5</span>)</span>
</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Practice addition with the following [mathjaxinline]2\times 2[/mathjaxinline] matrices: </p>
<table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\begin{pmatrix} -2 &amp; &amp; 1 \\ 1 &amp; &amp; -2 \end{pmatrix}+ \begin{pmatrix} 2 &amp; &amp; 1 \\ -2 &amp; &amp; 2 \end{pmatrix}=[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Note: The format for entering a matrix is the following (see the "Input Help" link for more): </p>
<table id="a0000000008" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\begin{pmatrix} 1 &amp; &amp; 2 \\ 3 &amp; &amp; 4 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
would be entered as: <code>[[1, 2],[3,4]]</code> </p>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<code>e, pi</code>
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enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Adding Bigger Matrices
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<p><b class="bfseries">(Part b)</b> Practice addition with the following [mathjaxinline]3\times 3[/mathjaxinline] matrices: </p>
<table id="a0000000012" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\begin{pmatrix} 1 &amp; &amp; 1 &amp; &amp; 0\\ 4 &amp; &amp; -2 &amp; &amp; 1 \\ 0 &amp; &amp; 3 &amp; &amp; 2 \end{pmatrix}+ \begin{pmatrix} 0 &amp; &amp; 3 &amp; &amp; 1\\ 1 &amp; &amp; -1 &amp; &amp; 2\\ 5 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}=[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
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<h2 class="hd hd-2 unit-title">L7Q2: Matrix Algebra Review: Multiplication</h2>
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Multiplying Matrices - Intro
</h3>
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<p>
<b class="bfseries">Product of a scalar and a matrix</b>
</p>
<p>
When a matrix is multiplied by a scalar, the same multiplication is carried out over all elements of the matrix: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\alpha \begin{pmatrix} a &amp; &amp; b \\ c &amp; &amp; d \end{pmatrix}= \begin{pmatrix} \alpha a &amp; &amp; \alpha b \\ \alpha c &amp; &amp; \alpha d \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>1</span>)</span>
</td>
</tr>
</table>
<p>
Properties of scalar multiplication include: </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax](\alpha + \beta )\textbf{A} = \alpha \textbf{A} + \beta \textbf{A}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>2</span>)</span>
</td>
</tr>
</table>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\alpha (\textbf{A} + \textbf{B}) = \alpha \textbf{A} + \alpha \textbf{B}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>3</span>)</span>
</td>
</tr>
</table>
<table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]1 \cdot \textbf{A} = \textbf{A}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>4</span>)</span>
</td>
</tr>
</table>
<table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]0 \cdot \textbf{A} = \textbf{0}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>5</span>)</span>
</td>
</tr>
</table>
<p>
where [mathjaxinline]\textbf{0}[/mathjaxinline] is the null matrix, defined as: </p>
<table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{0} = \begin{pmatrix} 0 &amp; &amp; 0 \\ 0 &amp; &amp; 0 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>6</span>)</span>
</td>
</tr>
</table>
<p>
<b class="bfseries">Product of two matrices</b>
</p>
<p>
When one matrix is multiplied by another matrix, we take the dot-product between each row of the first matrix and each column of the second matrix, resulting in a new matrix: </p>
<table id="a0000000008" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000009">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \begin{pmatrix} a &amp; &amp; b \\ c &amp; &amp; d \end{pmatrix}\begin{pmatrix} e &amp; &amp; f \\ g &amp; &amp; h \end{pmatrix}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \begin{pmatrix} \begin{pmatrix} a &amp; &amp; b \end{pmatrix}\cdot \begin{pmatrix} e \\ g \end{pmatrix}&amp; &amp; \begin{pmatrix} a &amp; &amp; b \end{pmatrix}\cdot \begin{pmatrix} f \\ h \end{pmatrix}\\ \begin{pmatrix} c &amp; &amp; d \end{pmatrix}\cdot \begin{pmatrix} e \\ g \end{pmatrix}&amp; &amp; \begin{pmatrix} c &amp; &amp; d \end{pmatrix}\cdot \begin{pmatrix} f \\ h \end{pmatrix}\end{pmatrix}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>7</span>)</span>
</td>
</tr>
<tr id="a0000000010">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \begin{pmatrix} (ae + bg) &amp; &amp; (af + bh) \\ (ce + dg) &amp; &amp; (cf + dh) \end{pmatrix}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>8</span>)</span>
</td>
</tr>
</table>
<p>
Properties of matrix products include: </p>
<table id="a0000000011" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}\textbf{B} \neq \textbf{B}\textbf{A}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>9</span>)</span>
</td>
</tr>
</table>
<p>
Note that there are special cases where [mathjaxinline]\textbf{A}\textbf{B} = \textbf{B}\textbf{A}[/mathjaxinline], for which matrices [mathjaxinline]\textbf{A}[/mathjaxinline] and [mathjaxinline]\textbf{B}[/mathjaxinline] are said to "commute." However, this commutation property is not generally true for any arbitrary pair of matrices. </p>
<table id="a0000000012" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax](\textbf{A}\textbf{B})\textbf{C} = \textbf{A}(\textbf{B}\textbf{C})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>10</span>)</span>
</td>
</tr>
</table>
<table id="a0000000013" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}(\textbf{B} + \textbf{C}) = \textbf{A}\textbf{B} + \textbf{A}\textbf{C}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>11</span>)</span>
</td>
</tr>
</table>
<table id="a0000000014" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{I} \cdot \textbf{A} = \textbf{A}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>12</span>)</span>
</td>
</tr>
</table>
<table id="a0000000015" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{0} \cdot \textbf{A} = \textbf{0}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>13</span>)</span>
</td>
</tr>
</table>
<p>
where [mathjaxinline]\textbf{0}[/mathjaxinline] was defined above and [mathjaxinline]\textbf{I}[/mathjaxinline] is the identity matrix, defined as: </p>
<table id="a0000000016" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{I} = \begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; 1 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>14</span>)</span>
</td>
</tr>
</table>
<p>
<b class="bfseries">Product of a matrix and a vector</b>
</p>
<p>
When a matrix is multiplied by a vector, we take the dot-product between each row of the matrix and the one column of the vector, resulting in a new vector: </p>
<table id="a0000000017" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000018">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \begin{pmatrix} a &amp; &amp; b \\ c &amp; &amp; d \end{pmatrix}\begin{pmatrix} e\\ f \end{pmatrix}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \begin{pmatrix} \begin{pmatrix} a &amp; &amp; b \end{pmatrix}\cdot \begin{pmatrix} e \\ f \end{pmatrix}\\ \begin{pmatrix} c &amp; &amp; d \end{pmatrix}\cdot \begin{pmatrix} e \\ f \end{pmatrix}\end{pmatrix}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>15</span>)</span>
</td>
</tr>
<tr id="a0000000019">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \begin{pmatrix} (ae + bf) \\ (ce + df) \end{pmatrix}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>16</span>)</span>
</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Practice scalar multiplication with the following [mathjaxinline]2\times 2[/mathjaxinline] matrices: </p>
<table id="a0000000020" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]4 \begin{pmatrix} -2 &amp; &amp; 1 \\ 1 &amp; &amp; -2 \end{pmatrix}- \begin{pmatrix} 0 &amp; &amp; -1 \\ 1 &amp; &amp; 2 \end{pmatrix}=[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
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<th class="formulainput" scope="col">Descriptions</th>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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Multiplying Matrices
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<p><b class="bfseries">(Part b)</b> Practice matrix multiplication with the following [mathjaxinline]2\times 2[/mathjaxinline] matrices: </p>
<table id="a0000000024" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\begin{pmatrix} -2 &amp; &amp; 1 \\ 1 &amp; &amp; -2 \end{pmatrix}\begin{pmatrix} 2 &amp; &amp; 1 \\ -2 &amp; &amp; 2 \end{pmatrix}=[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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Multiplying Bigger Matrices
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<p><b class="bfseries">(Part c)</b> Practice matrix multiplication with the following [mathjaxinline]3\times 3[/mathjaxinline] matrices: </p>
<table id="a0000000028" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\begin{pmatrix} 1 &amp; &amp; 1 &amp; &amp; 0\\ 4 &amp; &amp; -2 &amp; &amp; 1 \\ 0 &amp; &amp; 3 &amp; &amp; 2 \end{pmatrix}\begin{pmatrix} 0 &amp; &amp; 3 &amp; &amp; 1\\ 1 &amp; &amp; -1 &amp; &amp; 2\\ 5 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}=[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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<h2 class="hd hd-2 unit-title">L7Q3: Matrix Algebra Review: Determinants</h2>
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Matrix Determinants
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<p>
Consider the following [mathjaxinline]2\times 2[/mathjaxinline] matrix: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A} = \begin{pmatrix} a &amp; &amp; b \\ c &amp; &amp; d \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>1</span>)</span>
</td>
</tr>
</table>
<p>
The determinant, [mathjaxinline]\mathrm{det}(\textbf{A})[/mathjaxinline], also written as [mathjaxinline]|\textbf{A}|[/mathjaxinline], is given by the following: </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]|\textbf{A}| = ad - cb[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>2</span>)</span>
</td>
</tr>
</table>
<p>
For higher dimensional matrices, the determinant is calculated by taking "determinants of determinants" in the following way ( for [mathjaxinline]3\times 3[/mathjaxinline] matrix): </p>
<p>
Let, </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{B} = \begin{pmatrix} a &amp; &amp; b &amp; &amp; c\\ d &amp; &amp; e &amp; &amp; f\\ g &amp; &amp; h &amp; &amp; i \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>3</span>)</span>
</td>
</tr>
</table>
<p>
Then, </p>
<table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]|\textbf{B}| = a\left| \begin{pmatrix} e &amp; &amp; f \\ h &amp; &amp; i \end{pmatrix} \right| - b\left| \begin{pmatrix} d &amp; &amp; f \\ g &amp; &amp; i \end{pmatrix} \right| + c\left| \begin{pmatrix} d &amp; &amp; e \\ g &amp; &amp; h \end{pmatrix} \right|[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>4</span>)</span>
</td>
</tr>
</table>
<table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]|\textbf{B}| =a(ei-hf)-b(di-gf)+c(dh-ge)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>5</span>)</span>
</td>
</tr>
</table>
<p>
The following exercises provide practice with matrix determinants. </p>
<p><b class="bfseries">(Part a)</b> Find the determinant of [mathjaxinline]\textbf{A}[/mathjaxinline], where </p>
<table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A} = \begin{pmatrix} -2 &amp; &amp; 1 \\ 1 &amp; &amp; -2 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
<p style="display:inline">[mathjaxinline]|\textbf{A}| =[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Bigger Matrix Determinants
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<p><b class="bfseries">(Part b)</b> Find the determinant of [mathjaxinline]\textbf{B}[/mathjaxinline], where </p>
<table id="a0000000009" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{B} = \begin{pmatrix} 1 &amp; &amp; 1 &amp; &amp; 0\\ 4 &amp; &amp; -2 &amp; &amp; 1 \\ 0 &amp; &amp; 3 &amp; &amp; 2 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
<p style="display:inline">[mathjaxinline]|\textbf{B}| =[/mathjaxinline] </p>
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<code>2520</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L7Q4: Inverse Operations</h2>
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Inverting a Matrix
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<p>
The inverse of a matrix is defined in the following way: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}^{-1} = \frac{1}{|\textbf{A}|}\mathrm{adj}(\textbf{A})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>1</span>)</span>
</td>
</tr>
</table>
<p>
where [mathjaxinline]\mathrm{adj}(\textbf{A})[/mathjaxinline] is the adjoint of [mathjaxinline]\textbf{A}[/mathjaxinline]. </p>
<p>
Note that only square matrices have an inverse. </p>
<p>
We just learned about determinants, which we will use frequently. The adjoint is something we will not use frequently, but we provide a definition here, by way of example. </p>
<p>
Consider the following [mathjaxinline]2\times 2[/mathjaxinline] matrix: </p>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A} = \begin{pmatrix} a &amp; &amp; b \\ c &amp; &amp; d \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>2</span>)</span>
</td>
</tr>
</table>
<p>
The adjoint of the matrix is found by swapping the diagonal elements and inverting the sign of the off-diagonal elements: </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\mathrm{adj}(\textbf{A}) = \begin{pmatrix} d &amp; &amp; -b \\ -c &amp; &amp; a \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>3</span>)</span>
</td>
</tr>
</table>
<p>
Combining with the determinant, the inverse matrix is: </p>
<table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}^{-1} = \frac{1}{ad-cb} \begin{pmatrix} d &amp; &amp; -b \\ -c &amp; &amp; a \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>4</span>)</span>
</td>
</tr>
</table>
<p>
This is rather convoluted, but it is much neater when the matrix is "diagonal." Consider the diagonal matrix below: </p>
<table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{C} = \begin{pmatrix} a &amp; &amp; 0 \\ 0 &amp; &amp; d \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>5</span>)</span>
</td>
</tr>
</table>
<p>
The inverse of this matrix is: </p>
<table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{C}^{-1} = \begin{pmatrix} \frac{1}{a} &amp; &amp; 0 \\ 0 &amp; &amp; \frac{1}{d} \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>6</span>)</span>
</td>
</tr>
</table>
<p>
This property is generalizable to all square, diagonal matrices (not just [mathjaxinline]2\times 2[/mathjaxinline] matrices). Only square matrices have an inverse. </p>
<p>
Properties of matrix inverse: </p>
<table id="a0000000008" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}^{-1}\textbf{A} = \textbf{I}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>7</span>)</span>
</td>
</tr>
</table>
<table id="a0000000009" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}\textbf{A}^{-1} = \textbf{I}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>8</span>)</span>
</td>
</tr>
</table>
<table id="a0000000010" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax](\textbf{A}^{-1})^{-1} = \textbf{A}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>9</span>)</span>
</td>
</tr>
</table>
<table id="a0000000011" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax](\textbf{A}\textbf{B})^{-1} = \textbf{B}^{-1}\textbf{A}^{-1}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>10</span>)</span>
</td>
</tr>
</table>
<table id="a0000000012" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax](\alpha \textbf{A})^{-1} = \frac{1}{\alpha }\textbf{A}^{-1}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>11</span>)</span>
</td>
</tr>
</table>
<p>
where [mathjaxinline]\textbf{I}[/mathjaxinline] is the identity matrix, defined as: </p>
<table id="a0000000013" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{I} = \begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; 1 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>12</span>)</span>
</td>
</tr>
</table>
<p>
The following exercises provide practice with matrix inverse operations. </p>
<p><b class="bfseries">(Part a)</b> Find the inverse of [mathjaxinline]\textbf{A}[/mathjaxinline], where </p>
<table id="a0000000014" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A} = \begin{pmatrix} -2 &amp; &amp; 1 \\ 1 &amp; &amp; -2 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p style="display:inline">[mathjaxinline]\textbf{A}^{-1}=[/mathjaxinline]</p>
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<td class="formulainput">integers</td>
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<code>2520</code>
</td>
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<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Inverting a Bigger Matrix
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<p><b class="bfseries">(Part b)</b> Find the inverse of [mathjaxinline]\textbf{B}[/mathjaxinline], where </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\textbf{B} = \begin{pmatrix} 1 &amp; &amp; 0 &amp; &amp; 0\\ 0 &amp; &amp; -2 &amp; &amp; 0 \\ 0 &amp; &amp; 0 &amp; &amp; 2 \end{pmatrix}[/mathjax]</td>
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<p style="display:inline">[mathjaxinline]\textbf{B}^{-1}=[/mathjaxinline]</p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L7v1: Rewriting Equations of Motion Using Matrix Notation</h2>
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<h2 class="hd hd-2 unit-title">L7Q5: M and K matrices</h2>
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M and K Matrices
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As in the previous lesson, again consider the system of two coupled masses, attached to springs, constrained to move vertically like pistons. The masses are attached to each other by a massless string, which provides approximately constant tension in the direction of the string. </p>
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You previously derived equations of motion: </p>
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle m_1\ddot{y}_{1}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = -(k_{1} + \kappa )y_{1} + \kappa y_{2}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
<tr id="a0000000013">
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[mathjaxinline]\displaystyle m_2\ddot{y}_{2}[/mathjaxinline]
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[mathjaxinline]\displaystyle = -(k_{2} + \kappa )y_{2} + \kappa y_{1}[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<p>
Based on these equations of motion, identify the matrices [mathjaxinline]\textbf{M}[/mathjaxinline] and [mathjaxinline]\textbf{K}[/mathjaxinline]. Express your answers in terms of <code>m_1</code> for [mathjaxinline]m_{1}[/mathjaxinline], <code>m_2</code> for [mathjaxinline]m_{2}[/mathjaxinline], <code>k_1</code> for [mathjaxinline]k_{1}[/mathjaxinline], <code>k_2</code> for [mathjaxinline]k_{2}[/mathjaxinline], and <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline]. </p>
<p style="display:inline">[mathjaxinline]\textbf{M}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\textbf{K}=[/mathjaxinline]</p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<code>e, pi</code>
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<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<code>abs, ln, sqrt</code>
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<code>sin, cos, tan, sec, csc, cot</code>
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<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Understanding the Vectors and Matrices
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Consider the following questions related to the matrix equation [mathjaxinline]\left(\textbf{M}^{-1}\textbf{K} - \omega ^{2}\textbf{I}\right)\textbf{A} = 0[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> The vector [mathjaxinline]\textbf{A}[/mathjaxinline] represents: </p>
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<text> a) the location of each mass at any given time</text>
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<text> b) the amplitude of oscillation of each mass, which is not a time dependent quantity</text>
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<text> c) nothing really, it is a mathematical trick</text>
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Possible Solution - I
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<p><b class="bfseries">(Part b)</b> A possible solution to the equation [mathjaxinline]\left(\textbf{M}^{-1}\textbf{K} - \omega ^{2}\textbf{I}\right)\textbf{A} = 0[/mathjaxinline], is [mathjaxinline]\textbf{A}=0[/mathjaxinline] (True or False)? </p>
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<p><b class="bfseries">(Part c)</b> A possible way to solve the equation [mathjaxinline]\left(\textbf{M}^{-1}\textbf{K} - \omega ^{2}\textbf{I}\right)\textbf{A} = 0[/mathjaxinline], is to ensure the following determinant is equal to zero: [mathjaxinline]\left|\textbf{M}^{-1}\textbf{K} - \omega ^{2}\textbf{I}\right|=0[/mathjaxinline] (True or False)? </p>
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Multiplying the Matrices
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<p><b class="bfseries">(Part a)</b> Again consider the system of two coupled masses, attached to springs, constrained to move vertically like pistons. The masses are attached to each other by a massless string, which provides approximately constant tension in the direction of the string. </p>
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Identify the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix. Express your answers in terms of <code>m_1</code> for [mathjaxinline]m_{1}[/mathjaxinline], <code>m_2</code> for [mathjaxinline]m_{2}[/mathjaxinline], <code>k_1</code> for [mathjaxinline]k_{1}[/mathjaxinline], <code>k_2</code> for [mathjaxinline]k_{2}[/mathjaxinline], and <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline]. </p>
<p style="display:inline">[mathjaxinline]\textbf{M}^{-1}\textbf{K}=[/mathjaxinline]</p>
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<code>2520</code>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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<h3 class="hd hd-3 problem-header" id="w3_lect_07_01f-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@w3_lect_07_01f-problem-progress" tabindex="-1">
Finding Eigenfrequencies
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<p><b class="bfseries">(Part b)</b> To simplify things, let [mathjaxinline]m_{1} = m_{2} = m[/mathjaxinline] and [mathjaxinline]k_{1} = k_{2} = k[/mathjaxinline]. Now, solve for the eigenfrequencies of the system. To do so, find the solutions of the equation [mathjaxinline]|\textbf{M}^{-1}\textbf{K} - \omega ^{2}\textbf{I}|=0[/mathjaxinline], using the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix that you previously identified. </p>
<p>
Note, your answer will take the form of a quadratic equation with two solutions. Therefore, report the eigenfrequencies with [mathjaxinline]\omega _{+}[/mathjaxinline] indicating the larger of the two solutions and [mathjaxinline]\omega _{-}[/mathjaxinline] indicating the smaller one. Express your answer in terms of <code>m</code>, <code>k</code>, and <code>kappa</code> for [mathjaxinline]\kappa[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _{+} =[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\omega _{-} =[/mathjaxinline]</p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h3 class="hd hd-3 problem-header" id="w3_lect_07_01g-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@w3_lect_07_01g-problem-progress" tabindex="-1">
Frequency Limits
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<p><b class="bfseries">(Part c)</b> Now, consider the limits where the coupling (the string in this case) is either very strong ([mathjaxinline]\kappa \gg k[/mathjaxinline]) or very weak ([mathjaxinline]\kappa \ll k[/mathjaxinline]). What is the behavior of the system in each case? </p>
<p>
When the coupling is very strong, <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_w3_lect_07_01g_2_1">
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<input type="radio" name="input_w3_lect_07_01g_2_1" id="input_w3_lect_07_01g_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="w3_lect_07_01g_2_1-choice_1-label" for="input_w3_lect_07_01g_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_w3_lect_07_01g_2_1"> <text> a) the difference in normal mode frequencies is very small, and the two oscillators behave nearly completely independently.</text>
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<input type="radio" name="input_w3_lect_07_01g_2_1" id="input_w3_lect_07_01g_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="w3_lect_07_01g_2_1-choice_2-label" for="input_w3_lect_07_01g_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_w3_lect_07_01g_2_1"> <text> b) the difference in normal mode frequencies is very large and the motion of each oscillator depends strongly on the other.</text>
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<input type="radio" name="input_w3_lect_07_01g_2_1" id="input_w3_lect_07_01g_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="w3_lect_07_01g_2_1-choice_3-label" for="input_w3_lect_07_01g_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_w3_lect_07_01g_2_1"> <text> c) the behavior of the system does not depend on the strength of the coupling.</text>
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When the coupling is very weak, <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_w3_lect_07_01g_3_1">
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<input type="radio" name="input_w3_lect_07_01g_3_1" id="input_w3_lect_07_01g_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="w3_lect_07_01g_3_1-choice_1-label" for="input_w3_lect_07_01g_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_w3_lect_07_01g_3_1"> <text> a) the difference in normal mode frequencies is very small, and the two oscillators behave nearly completely independently.</text>
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